...and 27 posts so far without anyone mentioning that dreaded 4 letter word P***
Counting cadence - how about doing it indirectly by counting breathing instead? If you do something like breathe in with every other right footstrike (as in my case), it's a matter of multiplying by 4.
No-one so far has mentioned muscle elasticity...but if you spend less time in contact with the ground, that's going to increase your cadence by a few percent as well. An argument for some (simple) plyometric exercises perhaps?
Finally - just to be clear about the objective. Speed = cadence x stride length. You can try to increase either of the latter two in your running, but both come at a cost in terms of the energy you're expending per unit time. So as PP more or less says above, it's a matter of finding the appropriate combination of cadence and stride length that enables you to run at the same speed (at least) for the minimum energy expenditure. The OP appeared to be expending too much energy on stride length - presumably due to lead foot landing too far ahead of centre of mass - but you can also go too far the other way.